Ptolemaic negative type and the values q(6) and q(7)

Baker, Huh, Kummer, and Lorscheid define the triangular-hyperfield threshold q(n)=q(U_{2,n}) and conjecture exact values for all n. Using their identity q(n)=P(n-1), where P(m) is the universal negative-type exponent of m-point Ptolemaic metrics, we prove q(6)=\log_2(9/4) and q(7)=1. For five points, we separate zero-sum coefficient vectors by sign pattern. A sharp two-summand inequality proves the inequality for the 1+4 pattern. For the 2+3 pattern, a sharp four-point partial-correlation bound and an exact copositivity identity prove the required inequality. For six points, we construct an involutive coefficient transport under metric inversion that preserves the negative-type quadratic form. Applying the transport at an index with positive local contribution converts a hypothetical 3+3 counterexample into a 2+4 counterexample. Complete split graph metrics attain both bounds.